Solving nonlinear differential equations of Vanderpol, Rayleigh and Duffing by AGM

Abstract

In the present paper, three complicated nonlinear differential equations in the field of vibration, which are Vanderpol, Rayleigh and Duffing equations, have been analyzed and solved completely by Algebraic Method (AGM). Investigating this kind of equations is a very hard task to do and the obtained solution is not accurate and reliable. This issue will be emerged after comparing the achieved solutions by numerical method (Runge-Kutte 4th). Based on the comparisons which have been made between the gained solutions by AGM and numerical method, it is possible to indicate that AGM can be successfully applied for various differential equations particularly for difficult ones. The results reveal that this method is not only very effective and simple, but also reliable, and can be applied for other complicated nonlinear problems.

Key words： Algebraic Method (AGM); Angular Frequency; Vanderpol; Rayleigh; Duffing

Acknowledgements

The authors are grateful to the Ancient Persian mathematician, astronomer and geographer Muammadibn Musa Kharazmi, who was the one that his Compendious Book on Calculation by Completion and Balancing presented the first systematic solution of linear and quadratic equations. Furthermore, he was considered as the original inventor of algebra and is the one who Europeans derive the term ALGEBRA from his book and also the expression ALGORITHM has been taken from his name (the Latin form of his name). Consequently, we dedicate this way of solving linear and nonlinear differential equations to this scientist.

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